The Mysterious Story of Square Ice, Piles of Cubes, and Bijections

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MATHEMATICS Fig. 1. Families of objects counted by the same enumeration formula, for n = 3: ASMs; the six-vertex model with domain wall boundary conditions; square ice; fully packed loop conﬁgurations; classes of perfect matchings of the Aztec diamond graph; (not necessarily reduced) bumpless pipe dreams; monotone triangles with bottom row 12 ... n; DPPs; certain nonintersecting paths; cyclically symmetric lozenge tilings with a central hole; TSSCPPs; certain triangular shifted plane partitions; ASTs.

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TSSCPPs sign he of with and been alternating equivalence paper equinumerous ASMs has work, The ASM for classes 12. recent first while ref. very 13, symmetry (10) in in ref. of Zeilberger’s TSSCPPs in in study proved (not enumerated already the and was Andrews that graph, DPPs 1. including diamond for particular Fig. about Aztec statistics in of think the certain note to 6 of of ways to and matchings and equivalent 3 perfect ASMs other rows of for ASMs, that classes see to statistics Note configurations, equivalent dreams; certain model. loop is this pipe packed 1) study bumpless fully to Fig. reduced) model, physicists of necessarily by ice 2 developed square (row been the conditions had boundary are that wall ASMs “The techniques domain saying, the with in used by earlier) he problems decades opinion and several open matrices.” his physicists sign intriguing expresses by alternating he most introduced with been partitions do the to plane Eq. of have by on one partitions counted survey plane are is 254) what “This of p. bijections: question 8, the these (ref. concerns all Krattenthaler’s mystery finding unsolved, In still of proofs.” greatest, bijective problem of the the that area about optimistic the following are certain we the as but bijections. says DPPs, well such and as to Stanley TSSCPPs lead distribution between also joint as will same well paper as the this ASMs, have in plane and (https://www.mat.univie.ac.at/ ∼kratt/artikel/magog.html). that sketch TSSCPPs shifted conjecture statistics between we triangular gog-magog bijection by methods certain explicit A supported and an Krattenthaler, further TSSCPPs lack C. was still between and this We bijection dimensions 7–9, Again simple refs. 1. of a see Fig. is box of refinements; 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MATHEMATICS of other identities that are of interest to combinatorialists. Roughly speaking, the framework should be applicable for translating “computational” proofs of identities that involve only additions, subtractions, and multiplications (but not divisions), as detailed below. Note that it is probably more complicated to transfer ASM proofs using the six-vertex model approach (11), as such proofs typically employ an interpolation argument. 3. Signed Sets and Sijections It is widely accepted in combinatorics that bijective proofs of identities are “the best” in most circumstances: They typically bring the most clarity to a statement, they yield interesting generalizations, and they are usually esthetically pleasing. For example, the Pn n n n statement k=0 k = 2 can be proved in a variety of ways, e.g., by induction, or by finding the expansion of (1 + x) and plugging in x = 1. On the other hand, a bijective proof of this statement is the simple observation that the right-hand side counts all subsets of an n-element set, while the left-hand side splits them according to size. Pn k n Things are a little different when the identity involves signs. For example, consider the identity k=0(−1) k = 0 for n ∈ Z>0. In this case, a “bijective” proof means that we find the right cancellations: We have to cancel a set of even size with a set of odd size. For example, we could map a set A to A \{n} or A ∪ {n}, depending on whether or not n ∈ A. This map has the added benefit of Pm k n m n−1 proving k=0(−1) k = (−1) m at no extra cost. Since the identity Eq. 2 has no signs, a proof that would avoid signs and cancellations would be preferable. Our proof, however, uses them quite substantially. This stems from the fact that this proof has been developed from the nonbijective proof by I.F. which contains calculations that involve signs. This also raises the question of whether a possible bijective proof that avoids signs can in turn be translated into a computational proof that avoids signs. No such nonbijective proof is currently known. In the remainder of this section, we briefly introduce the concepts of signed sets and sijections, signed bijections between signed sets. We present the basic concepts here and refer the reader to ref. 29, section 2 for all of the details and more examples. A signed set is a pair of disjoint finite sets: S = (S +, S −) with S + ∩ S − = ∅. Equivalently, a signed set is a finite set S together with a sign function sign: S → {1, −1}. Signed sets are usually underlined throughout this paper with the following exception: An ordinary set S always induces a signed set S = (S, ∅), and in this case we identify S with S. We summarize related notions. The size of a signed set S is |S| = |S +| − |S −|. The opposite signed set of S is

− S = (S −, S +). [3]

The Cartesian product of signed sets S and T is S × T = (S + × T + ∪ S − × T −, S + × T − ∪ S − × T +). The disjoint union of signed sets S and T is S t T = (S × ({0}, ∅)) ∪ (T × ({1}, ∅)). These constructions correspond as usual to arithmetic operations on the sizes; i.e., |S × T | = |S| · |T | and |S t T | = |S| + |T |. [4]

The disjoint union of a family of signed sets S t indexed with a signed set T is G [ S t = (S t × {t}). t∈T t∈T

Here {t} is ({t}, ∅) if t ∈ T + and (∅, {t}) if t ∈ T −. Most of the usual properties of Cartesian products and disjoint unions (commutativity, distributivity, etc.) of ordinary sets extend to signed sets. An important type of signed sets is signed intervals: For a, b ∈ Z, deﬁne ( ([a, b], ∅) if a ≤ b [a, b] = . (∅, [b + 1, a − 1]) if a > b

Here [a, b] stands for the usual interval in Z, deﬁned when a ≤ b. Note that we always have |[a, b]| = b − a + 1. The signed sets that are of relevance in this paper are usually constructed from signed intervals using Cartesian products and disjoint unions. The role of bijections for signed sets is played by “signed bijections,” which we call sijections, and they are manifestations of the fact that two signed sets have the same size. A sijection ϕ from S to T ,

ϕ: S =⇒ T ,

is an involution on the set (S + ∪ S −) t (T + ∪ T −) with the property ϕ(S + t T −) = S − t T +. It follows that also ϕ(S − t T +) = S + t T −. A sijection can also be thought of as a collection of a sign-reversing involution on a subset of S, a sign-reversing involution on a subset of T , and a sign-preserving matching between the remaining elements of S with the remaining elements of T . The existence of a sijection ϕ: S ⇒ T clearly implies |S| = |S +| − |S −| = |T +| − |T −| = |T |. In Fig. 2, Left the sijection is a bijection between the blue (resp. green) parts of S + and S − (resp. T + and T −) and between the light gray (resp. dark gray) parts of S + and T + (resp. S − and T −). A sijection between two signed sets with no negative elements is clearly a bijection. Our two bijections are constructed from two chains of sijections with several intermediate sets connecting the two pairs of sets for which we want to show equinumerosity. However, to be able to use these sijections to construct the two bijections, we need a notion of composing sijections. While composing bijections is of course trivial, this turns out to be slightly more complicated for general sijections. There seems to be only one natural choice for how to do this; indeed, the construction is a generalization of the Garsia–Milne involution principle. For an illustration of this, see Fig. 2, Right. There we have a sijection ϕ between S and T (solid lines) and a sijection ψ between T and U (dashed lines); through ϕ (resp. ψ), we have a bijection between the blue (resp. green) parts of S (resp. U ), and all other elements of S + or U − are mapped to a unique element of S − or U + via an alternating sequence of applications of ϕ and ψ. For the formal deﬁnition of composition as well as of the Cartesian product and the disjoint union of sijections see proposition 2 of ref. 29.

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GT the us what shifted for allows in formula it operator patterns abovementioned GT but the shifted row, of bottom proof sijective fixed a a is alone That patterns). GT c +1 ] → 1 or , . . . [a A ≤n A , i , b ,j i n ,j k ] > n t n lota hsnme equals number this that also and stesto l Sso size of ASMs all of set the is > ≤n [c ) A [ ∈ b A 1, + i h ieto svr ipe u ed aet pi tit ae.If cases. into it split to have do we but simple, very is sijection The n DPP and , 1, + Z ,j i −1,j +1 n en recursively define , b c seven. is te ae r similar. are cases Other ]. > If ]. A i a ,j n Given ,i +1 ≤ stesbe ftoewt exactly with those of subset the is c i.2. Fig. hnvraltretrsaedfie,adtesg faG atr s1i n nyi h ubrof number the if only and if 1 is pattern GT a of sign the and defined, are terms three all whenever GT < b a α (k) = (k) then , , = b lutaino ieto (Left sijection a of Illustration , satinua array triangular a as α c a ∈ GT n ,b [a DPP Z, ,c × , (k [ : b n osrc sijection a construct Q [ = ] 1 a ASM , n , j n , −1 . . . =0 c −1 a ] × , = , c (3j k ( ⇒ n ASM ] n n ,i ∪ +j +1)! = ) [ a stesbe ftoewt nposition in 1 with those of subset the is [c )! , n l∈[k b 1, + hr r ormi tp ftepof h rtse sdsrbdin described is step first The proof. the of steps main four are There . A ,i ] t −→ 1 (A = n fcmoiino ietos(Right sijections of composition of and ) ,k [b b i 2 ] k − ] 1, + ×···×[k ASM and ∈ i 1 G ,j en GT define Z, curne of occurrences ) 1≤j c [ b n ] n −1 [a = 1, + −1 ≤i × ,k ≤n , n c DPP b ] ] GT ] of [ (∅, = − t (l n (k n ,i 1 [c +1 2 n ) , , c h ytmo iereutoscnas be also can equations linear of system The . . . . 1, + ohv igepstv lmn,adfor and element, positive single a have to 1, + ubr,weew have we where numbers, a , l n ≤ b n b −1 ] × b . ow ipycne h w copies two the cancel simply we so ]), ≤ n ). [P (1, ). c qastenme fDP with DPPs of number the equals ij etk h aua bijection natural the take we , ] i i m DPP ), ,j =1 fsge esa signed a as sets signed of NSLts Articles Latest PNAS n 4). stesto l DPPs all of set the is A i ,j ≤ A i | −1,j f7 of 5 [5] ≤

MATHEMATICS DPPn−1 × Bn,1 × ASMn,i −→ DPPn−1 × ASMn,1 × Bn,i , [6]

where Bn,i is the set of (2n − 1) subsets of {1, 2, ... , 3n − 2} with median n + i − 1. It can readily be checked that this serves as a bijective proof of Eq. 2. Note that Eqs. 5 and 6 involve seemingly unnecessary factors, which cancel when taking cardinalities. On the level of bijections it is somewhat more natural to keep these factors because division cannot be mimicked as naturally as the three other basic arithmetic operations (Eqs. 3 and 4) by a construction for signed sets. A method we use several times in step 1 is to use disjoint unions and Cartesian products of the sijection α to construct some sijections for disjoint unions of signed boxes (Cartesian products of signed intervals), then to use disjoint unions of those to construct sijections for disjoint unions of GT patterns, and then to use those to construct sijections for monotone triangles. As an example, let us sketch one such construction.

n−1 n−1 Problem 2 (ref. 29, problem 2). Given a = (a1, ... , an−1) ∈ Z , b = (b1, ... , bn−1) ∈ Z , x ∈ Z, write S i = ({ai }, ∅) t (∅, {bi + 1}), and construct a sijection

G β = βa,b,x :[a1, b1] × · · · × [an−1, bn−1] =⇒ [l1, l2] × [l2, l3] × · · · × [ln−2, ln−1] × [ln−1, x].

(l1,...,ln−1)∈S 1×···×S n−1

Construction: The case n = 2 is constructed in Problem 1. For n ≥ 3, we get, by induction, a sijection to

    G G [a1, b1] × [a2, l3] × · · · × [ln−1, x]t [a1, b1] × (−[b2 + 1, l3]) × · · · × [ln−1, x],

(l3,...,ln−1)∈S 3×···×S n−1 (l3,...,ln−1)∈S 3×···×S n−1

and then we use sijections α from [a1, b1] to [a1, a2] t (−[b1 + 1, a2]) and [a1, b2 + 1] t (−[b1 + 1, b2 + 1]), respectively.

n−1 n−1 Problem 3 (ref. 29, problem 4). Given a = (a1, ... , an−1) ∈ Z , b = (b1, ... , bn−1) ∈ Z , x ∈ Z, construct a sijection G G ρ = ρa,b,x : GT(l) =⇒ GT(l1, ... , ln−1, x),

l∈[a1,b1]×···×[an−1,bn−1] (l1,...,ln−1)∈S 1×···×S n−1

where S i = ({ai }, ∅) t (∅, {bi + 1}). Construction: Take a disjoint union (properly deﬁned) of sijections β, and we obtain a sijection

G G GT(m) =⇒ GT(m). F m∈[a1,b1]×···×[a ,b ] m∈ [l1,l2]×[l2,l3]×···×[l ,l ]×[l ,x] n−1 n−1 (l1,...,ln−1)∈S1×···×Sn−1 n−2 n−1 n−1

By basic constructions, we get a sijection to F F GT(m), and by deﬁnition of (l1,...,ln−1)∈S 1×···×S n−1 m∈[l1,l2]×···×[ln−2,ln−1]×[ln−1,x] F GT, this is equal to GT(l1, ... , ln−1, x). (l1,...,ln−1)∈S 1×···×S n−1 After several such results concerning the signed sets of GT patterns, we can prove that the signed set of shifted GT patterns, denoted by SGT(k), satisﬁes the same recursive identity as the signed set of generalized monotone triangles MT(k). For monotone triangles with a strictly increasing bottom row, the recursion can be understood quite easily: If we delete the bottom row, say, k = (k1, ... , kn ) of a monotone triangle, then we obtain a monotone triangle with a new bottom row, say, l = (l1, ... , ln−1) where k1 ≤ l1 ≤ k2 ≤ l2 ≤ ... ≤ ln−1 ≤ kn and l1 < l2 < . . . < ln−1. It is also possible to write the resulting recursion more conveniently as a disjoint union over signed boxes. For n = 3, this would be

  G G G MT(k1, k2, k3) = MT(l1, l2) t MT(l1, l2) t − MT(l1, l2). (l1,l2)∈[k1,k2−1]×[k2,k3] (l1,l2)∈[k1,k2]×[k2+1,k3] (l1,l2)∈[k1,k2−1]×[k2+1,k3]

The difﬁcult part is to show that the shifted GT patterns satisfy the same recursive identity (29); the construction uses many previously constructed sijections such as ρ. We omit the details here due to space limitations, but the resulting sijection is of the form

G G Φ = Φk,x : SGT(l) =⇒ SGT(k).

µ∈ARn l∈e(k,µ)

Here ARn is a certain (simple) signed set of arrow rows, and e(k, µ) is a certain deformation of k. Together with the sijection that proves the same recursion for generalized monotone triangles, we obtain a sijection

Γ = Γk,x : MT(k) =⇒ SGT(k) [7]

by induction.

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